Simple Representations of Zero-Net Mass-Flux Jets in Grazing Flow for Flow-Control Simulations

Transcription

1 Simple Representations of Zero-Net Mass-Flux ets in Grazing Flow for Flow-Control Simulations by Ehsan Aram, Rajat Mittal an Louis Cattafesta Reprinte from International ournal of Flow Control olume 2 Number 2 une 2 Multi-Science Publishing ISSN

2 9 Simple Representations of Zero-Net Mass-Flux ets in Grazing Flow for Flow-Control Simulations Ehsan Aram, Rajat Mittal 2 Department of Mechanical Engineering, ohns Hopkins University, Baltimore, MD an Louis Cattafesta 3 Interisciplinary Microsystems Group Department of Mechanical an Aerospace Engineering, University of Floria, Gainesville, FL Simple bounary conitions that can represent the flow emanating from zero-net massflux (ZNMF) jets in grazing flows are examine. Two-imensional Navier-Stokes simulations of ZNMF jets in grazing flow are use to etermine the key characteristics of the jet profile, an these are use to construct a series of bounary conition moels. These bounary conition moels are then teste for a jet emanating into an attache bounary layer as well as a bounary layer with an inuce separation bubble. A key fining of the current stuy is that simple plug-flow type moels of ZNMF jets can lea to incorrect trens in separation control. Base on this analysis, we select two moels that are simple, yet provie a reasonable representation of the ZNMF jet in a grazing flow. An empirical closure of these moels is also suggeste. NOMENCLATURE C uu = u momentum flux in x-irection C uv = u momentum flux in y-irection _ C uv = Time average of C uv C vu = v momentum flux in x-irection C vv = v momentum flux in y-irection _ C vv = Time average of C vv C vvv = ertical kinetic energy flux in y-irection C vvv = Time average of Cvvv = Slot with F + = Dimensionless forcing frequency, f /f Sep f = Forcing frequency of ZNMF jet f Sep = Separation bubble frequency H = Cavity height h = Slot height Q = olume flow rate of ZNMF jet Q = olume flow rate amplitue of ZNMF jet Re = Bounary layer Reynols number, U v Re = et Reynols number, /v S = Stokes number, ω 2 / v Grauate Stuent, Department of Mechanical Engineering, earam@jhu.eu 2 Professor, Department of Mechanical Engineering, mittal@jhu.eu 3 Professor, Department of Mechanical & Aerospace Engineering, cattafes@ufl.eu olume 2 Number 2 2

3 Simple Representations of Zero-Net Mass-Flux ets in Grazing Flow for Flow-Control Simulations St = Strouhal number, 2π f / U i = Fourier expansion amplitues of u-velocity along the slot exit, i =,, 2, U = Time average of U over the expulsion phase U = Free stream velocity u = Streamwise velocity (in x-irection) u = Time average of u-velocity = Spatial average of jet v-velocity along the slot exit = Time average of uring the expulsion phase L = Spatial average of jet v-velocity along left half of the slot exit L = Time average of L over the expulsion phase R = Spatial average of jet v-velocity along right half of the slot exit _ R = Time average of R over the expulsion phase (2) i = Fourier expansion amplitues of L, i =,, 2, i = Fourier expansion amplitues of R, i =,, 2, v = Cross-stream velocity (in y-irection) W = Cavity with x = Horizontal irection y = ertical irection δ = Bounary layer thickness ξ z = Spanwise vorticity Ω = Spanwise vorticity flux Ω = Time average of Ω v = Kinematic viscosity ω = Angular forcing frequency of the ZNMF jet, 2p f I. INTRODUCTION Zero-net mass-flux (ZNMF) or synthetic jet actuators are versatile evices with numerous applications incluing active control of flow separation, mixing enhancement, heat transfer, an thrust vectoring of jets. -4 A number of parametric stuies inicate that several geometrical an flow parameters govern the performance characteristics of ZNMF jets an their control effectiveness. 5-7 The scales of these jets may be 2 3 times smaller in size compare to the ominant length scale of the problem (such as the wing chor). Thus, conucting numerical flow simulations which inclue full representations of the actuator cavity an/or slot is an extremely aunting proposition. 8 A more attractive approach is to evise a simpler representation of the actuator an use that instea of simulating the full actuator. A variety of such approaches have been use in the past. For instance, Kral et al. 9 simulate the jet flow associate with a two-imensional, incompressible synthetic jet actuator in the quiescent external flow by prescribing moifie bounary conitions for velocity an pressure at the jet exit. Comparison between the computational results an experimental ata for both steay an unsteay conitions showe goo agreement in the turbulent case, but the moel was not able to capture the breakown of the vortex train in the laminar case. Rizzetta et al. use the recore jet exit velocity profile of an isolate actuator as the bounary conition for the 2D an 3D external flow fiel. Other more sophisticate approaches have also been use with varying egrees of success. Filz et al. have moele 2D isolate synthetic jets as well as a synthetic jet in grazing flow using eterministic source terms traine by a neural network. The source terms were calculate from the time-average of the unsteay solution. This technique was able to reuce the computational time significantly an preict the momentum thickness of the flow with reasonable accuracy. A lowimensional moel of 2D an 3D synthetic jet actuator base on the unsteay compressible quasi-oneimensional Euler equations was propose by Yamaleev an Carpenter 2 an teste in grazing flow. Their moel satisfie the conservation laws an was able to preict the resonance characteristics of the actuator. Although this moel reuce the computational cost by eliminating the cavity, it i not consier the effects associate with the vortex generation, bounary layer, an the bounary-layer separation insie the actuator cavity. International ournal of Flow Control

4 Ehsan Aram, Rajat Mittal, Louis Cattafesta With regars to simplifie actuator moels, Rathnasingham an Breuer 3 evelope a system of first-orer, non-linear ifferential equations which represente the flui structure-interaction characteristics of the evice in a quiescent external flow. The membrane was moele as a circular plate, an the Bernoulli equation was use for flow in the slot. Comparison with experimental ata showe that the semi-empirical moel coul accurately preict the time an frequency characteristics of the jet. Gallas et al. 5 an Agashe et al. 4 have evelope lumpe-element moels (LEM) of piezoelectric an electroynamic actuators in the quiescent external flow, respectively. In this approach, each component of the actuator is moele as an element of an equivalent electrical circuit. Comparisons with the experimental results of prototype actuators reveal goo agreement, but this metho is incapable of proviing etaile profiles of the flow quantities at the slot/orifice exit. Motivate by the LEM, Ravi 5 use numerical simulations to explore the iea of replacing the ZNMF jet in grazing flow with appropriately selecte bounary conitions applie on the surface. He compare the simpler moel with full simulations that inclue the cavity an slot. Several caniate profiles were consiere. The overall conclusion was that none of these simple moels were able to prouce the effect of the jet on the cross flow bounary layer. Tang et al. 6 have recently preicte the instantaneous spatially average velocity at the orifice exit of a synthetic jet in quiescent flow by three ifferent moels which they call the ynamic incompressible flow moel, the static compressible flow moel an the (traitional) LEM. Comparisons with the experimental an numerical results suggeste that the LEM moel performs more reliably than the static compressible flow moel in ifferent conitions. A simple slot only moel base on incluing only the jet slot (an not the cavity) of the actuator in grazing flow has also recently been propose by Raju et al. 8 Comparison with the full cavity an a simple sinusoial plug flow velocity moel showe that this simplifie actuator moel was able to capture more of the flow physics associate with the full cavity simulations an coul more accurately preict the reuction of the separation bubble size for a inuce flow separation. Although the slot-only moel represents a significant reuction in the computational complexity require to represent the actuator in flow simulations, it still requires simulation of the unsteay viscous flow in the slot. Furthermore, if the slot location is varie, a new gri may have to be generate. If, however, the entire actuator can be replace by an appropriate an relatively simple velocity bounary conition at the jet exit, there woul be no nee to inclue a slot, an that woul significantly simplify the task of ZNMF flow control simulations. In the current paper, we explore the possibility of eveloping just such a bounary conition representation for ZNMF jets in grazing external flow. II. MODIFIED BOUNDARY CONDITION ACTUATOR MODELS The approach taken in the current stuy is to use the jet exit profile of a full cavity simulation (i.e. a simulation where the jet actuator cavity an slot are explicitly inclue) to construct a simple low-orer escription of the ZNMF jet. An attempt is mae to reprouce the flow physics associate with the full cavity simulations using our propose moels. Since in the current stuy the oscillation frequency of the iaphragm is consiere well below the Helmholtz frequency of the actuator, the incompressible flow assumption is fairly accurate insie the actuator an flow rate at the slot exit is irectly proportional to the volume isplacement of the iaphragm or piston that rives the flow. Utturkar et al. 7 showe that in the case of incompressible flow, the cavity shape has a very limite effect on the jet exit flow an we therefore assume a simple geometric moel for the actuator. In cases where compressibility effects insie the cavity cannot be ignore, the volume flux is certainly affecte by the cavity shape but can be preicte by lumpe-element type moels. 5, 4 However, regarless of whether the flow insie the cavity behaves in a compressible or incompressible manner, the etails of the jet profile an the momentum/energy flux from the jet, for a given jet volume flux, epen primarily on characteristics of the flow through the slot as well as the grazing flow. The current effort is irecte towars moeling these effects. Figure (a) & (b) show a schematic of the two configurations consiere in the current stuy: the Full Cavity (FC) an the Moifie Bounary Conition (MBC) moels. As it is seen in Fig. (a), the actuator iaphragm is locate at the bottom of the cavity, an a sinusoial plug flow profile on the bottom wall is prescribe to moel the effect of iaphragm motion on the flow. In the current stuy, the FC moel is consiere as the gol stanar, an the objective of the current stuy is to assess a variety of MBC moels by comparing with the FC moel. By far the simplest MBC moel is one that assumes a sinusoially varying plug flow for the vertical velocity profile at the jet exit uring the expulsion an ingestion phase. Numerical simulations also olume 2 Number 2 2

5 2 Simple Representations of Zero-Net Mass-Flux ets in Grazing Flow for Flow-Control Simulations v u u( x, t) v( x, t) y x h H W (a) (b) Figure : Schematic of (a) full cavity (FC) an (b) moifie bounary conition (MBC) moel. Figure 2: elocity vectors an instantaneous spanwise vorticity contours for the full cavity (top row) an MBC (bottom row) moels at four ifferent phases ϕ =, 9, 8, 27. The plots are for a case with Re = 25, S = 7.7, St =.2, U / = 4 an δ/ = 3. require the specification of the horizontal velocity component in this simplest MBC moel. Our past stuies (Raju et al. 8 ) have shown that, assuming a zero Neumann bounary conition for the horizontal velocity u uring the ingestion phase is essential for numerical stability, an so that is the conition employe for the horizontal velocity component uring the ingestion phase. Generally, this simple moel assumes that the horizontal velocity is zero uring the expulsion phase an the amplitue of the vertical velocity is chosen so as to match the volume flux of the FC moel. We enote this moel as MBC. In Fig. 2, we compare the instantaneous spanwise vorticity contours prouce by the FC moel to that of the MBC moel for a case where the jet is emanating into a grazing laminar bounary layer. The imensionless flow parameters for this comparison are Re = 25, St =.2, U / = 4 an δ/ = 3. A comparison of the external flow an the jet exit velocity profile suggests the following. First, the simple MBC type moel prouces an external flow that is ifferent from that of the FC moel, especially uring the expulsion phase. Secon, the vertical velocity shows significant variations along the slot with. Thir, the flow at the slot exit has a non-zero horizontal velocity component which coul potentially have an important effect on the jet. We use these observations as a basis for exploring a set of MBC moels in the following section. Two key factors nee to be consiere while esigning viable MBC moels for ZNMF jets in grazing flows. First, the moels shoul be as simple as possible an have a relatively small number of parameters. Secon, the moel shoul provie a reasonably goo approximation to the effects of the actual ZNMF jet. Keeping this in min, we introuce moels which have at most two spatial egrees International ournal of Flow Control

6 Ehsan Aram, Rajat Mittal, Louis Cattafesta 3 j x x 2 x 3 x 4 x Figure 3: Assume vertical velocity profile for MBC moels. of freeom for the vertical velocity. We prescribe the vertical velocity for at most two interior points across the slot exit. This allows us to prescribe a non-uniform velocity profile across the slot. Furthermore at any given time, we assume a linear velocity variation between these two points an impose the no-slip bounary conition at the slot eges. Figure 3 shows the type of vertical velocity profile that is employe in our MBC moels where (2) an are the points where the velocity is specifie an points () an (4) are the slot ege points. The points (2) an can in principle be locate anywhere, but we choose a simple prescription whereby the points are locate at 25% an 75% locations along the slot exit. With regars to the horizontal velocity component, in the interest of simplicity, we assume that there is no spatial variation of this component of velocity uring the expulsion phase. Note that the no-slip bounary conition is impose at the slot eges for horizontal velocity uring the expulsion an ingestion phases. We enote the vertical velocity at the points (2) an as (2) an, respectively, an the horizontal velocity as U. The next consieration is the time variation of this assume profile for (2), an. We first assume that the volume flux through the slot is sinusoial, i.e. Q(t) = Q sin(ωt) () Given this, it is natural to assume that the velocity at any given point across the jet slot is also perioic an can be expresse in terms of a Fourier series in time as follows: (2) (t) = (2) + (2) sin(ωt + ϕ (2) ) + (2) sin(2ωt + ϕ (2) 2 ) + (t) = + sin(ωt + ϕ ) + 2 sin(2ωt + ϕ 2 ) + (2) U (t) = U + U sin(ωt + ψ ) + U 2 sin(2ωt + ψ 2 ) + Base on conservation of mass we obtain the following relationships for the above coefficients: (2) = = We now look at the FC moel for guiance regaring the amplitues an phases of the terms in the above Fourier series. Figure 4(a) shows the velocity average over the right an left halves of the slot exit as well as the average velocity over the entire slot (which is an exact sinusoi). These are compute as: / 2 R () t = 2 L 2 () t = an () t = (4) vxtx (,) vxtx (,) ; vxtx (,) ; / 2 If we assume that (2) an are relate to L an R respectively, we fin from Fig. 4(a) that (2) an vary mostly at the ZNMF jet frequency implying that higher harmonics are negligible, i.e. (2) >> n (2) for n > (5a) >> n for n > (5b) olume 2 Number 2 2

7 4 Simple Representations of Zero-Net Mass-Flux ets in Grazing Flow for Flow-Control Simulations Secon, the two velocities are also approximately in phase with the volume flux, suggesting that φ (2), φ (6) Next, Fig. 4(b) shows the spatial average of the horizontal u velocity component at the exit of the full cavity over one cycle. The spatial integral is calculate over the whole length of the exit. Two observations can be mae from this plot. First, the time-space average horizontal velocity is non-zero. Secon, the horizontal velocity component possesses a large Fourier amplitue at the first super harmonic of the jet frequency, i.e. 2ω with ψ 2. Given all of the above observations, the following reuce representation of the velocity components can be obtaine for the two interior points in the slot: (2) (t) = + (2) sin(ωt) (t) = + sin(ωt) (7) U (t) = U + U 2 sin(2ωt) v/ j L 2 R φ φ (a) (b) Figure 4: The spatial average of the jet velocity component along the slot exit of the full cavity moel, (a) vertical v component of velocity, (b) horizontal u component of velocity. u/u Note that the above prescription has five coefficients. However, since (2) an are relate to the amplitue of the volume flux Q over the expulsion phase, only four of these coefficients are unknown. In orer to assess the viability of the above prescription (as well as further simplifications to it) we have evaluate the four remaining coefficients from the FC moel. This is one by extracting the velocity components at the slot exit as a function of space an time from the FC simulations, evaluating the spatial average of velocity along the slot exit for horizontal velocity an along each half of slot separately for vertical velocity an then subjecting these to a Fourier analysis. For instance, (2) is evaluate as the spatial average of the jet exit velocity on the left half of the slot, whereas is the corresponing velocity for the right half of the slot. We then create a number of variations of the above moel by making further simplifying assumption(s) an compute the flows for these simpler moels. The compute flows are then compare with the FC moel to assess the performance of each moel. Propose Moels Two key consierations in moel evelopment are fielity an complexity. Moel complexity is usually irectly relate to the number of unknowns in the moel that require closure. While the simplest moels usually ten to be the ones with the lowest fielity, increase in complexity beyon a certain stage is not necessarily accompanie by a corresponing increase in fielity. The objective of the exercise below is to ientify moels that are relatively simple, yet provie significant improvement in fielity over the simplest plug-flow moel. It shoul be pointe out that in general, expulsion an ingestion nee to be treate separately in these moels. This is because expulsion an ingestion for the jet represent inflow an outflow respectively for the computational omain, an numerical stability requires ifferent treatments for these two International ournal of Flow Control

8 Ehsan Aram, Rajat Mittal, Louis Cattafesta 5 Table : Different moifie bounary conition (MBC) moels examine in current stuy. Moel Expulsion Ingestion Unknowns Remarks Plug flow moel. U (t) =. u / y = Zero u-vel uring the (2) (t) = (2) sin(ωt) (2) (t) = (2) sin(ωt) expulsion phase. MBC (t) = sin(ωt) (t) = sin(ωt) None Uniform v-vel. uring (2) = = Q / (2) = = Q / the expulsion an ingestion phases. Simplest moel. u-vel same as MBC. Nonuniform v-vel U (t) =. u / y = amplitue with non MBC2 (2) (t) = + (2) sin(ωt) (2) (t) = + (2) sin(ωt) zero mean value in left (t) = + sin(ωt) (t) = + ( 3) sin(ωt) an right half of slot (2) (2) ( 2) (phase ifference) uring the expulsion an ingestion phases. U Sinusoial u-vel. with U (t) = U + U 2 sin(2ωt) u / y = U 2 non zero mean value in (2) (t) = + (2) sin(ωt) (2) (t) = + (2) sin(ωt) expulsion phase. MBC3 (t) = + sin(ωt) (t) = + sin(ωt) ( 3) v-vel. same as MBC2. (2) (2) ( 2) Most general moel. u-vel. same as MBC3. v-vel. same as MBC3 U (t) = U + U 2 sin(2ωt) u / y = U uring the expulsion (2) (t) = + (2) sin(ωt) (2) (t) = + (2) sin(ωt) U 2 phase. MBC4 (t) = + sin(ωt) (t) = + sin(ωt) Uniform v-vel. amplitue (2) (2) = = Q / ( 3) with non zero mean value in left an right half of slot uring the ingestion phase. u-vel. same as MBC4. v-vel. same as MBC4 U (t) = U + U 2 sin(2ωt) u / y = U with zero mean value (2) (t) = (2) sin(ωt) (2) (t) = (2) sin(ωt) U 2 in both halves of slot MBC5 (t) = sin(ωt) (t) = sin(ωt) ( 3) uring the expulsion (2) (2) = = Q / ( 2) phase. v-vel. same as MBC uring ingestion phase. U (t) = U u / y = U (2) (t) = (2) sin(ωt) (2) (t) = (2) sin(ωt) Constant u-vel. in MBC6 (t) = sin(ωt) (t) = sin(ωt) ( 3) expulsion phase. (2) (2) = = Q / ( 2) v-vel. same as MBC5. U (t) =. u / y = MBC7 (2) (t) = (2) sin(ωt) (2) (t) = (2) ( 3) sin(ωt) u-vel. same as MBC. (t) = sin(ωt) (t) = ( 2) sin(ωt) v-vel. same as MBC6. (2) (2) = = Q / ( 2) olume 2 Number 2 2

9 6 Simple Representations of Zero-Net Mass-Flux ets in Grazing Flow for Flow-Control Simulations situations. In particular, Dirichlet bounary conitions on all components of velocity cannot be impose at an outflow bounary in regions where there is significant flux of vorticity since oing so leas to numerical instability. An acceptable treatment uring ingestion therefore is to impose Dirichlet bounary conitions on the vertical component of velocity an homogeneous Neumann conitions on the horizontal component. Table lists the propose moels stuie in this paper. We start with the simplest moel (MBC) an improve the moel by aing the nonuniformity to the vertical velocity over the entire cycle (MBC2) an also prescribing the time-epenent horizontal velocity in expulsion phase in MBC3 (The most general an complex moel), an for the rest of moels, we try to simplify the MBC3 by removing unknowns an examine to what extent this simplification can influence the fielity of the results. Note that for MBC5-7, a uniform vertical velocity profile is use uring the ingestion phase. Note that in the above profiles, if (2) = then we have the so-calle plug flow profile wherein the vertical velocity is uniform across the slot exit at all time instants. Conversely, for moels with (2), we assume a non-uniform exit velocity profile. For moels with = we are assuming that both halves of the jet slot separately satisfy the zero-net mass-flux constraint. The simulations to stuy the performance of the ifferent moels in the vicinity of the actuator have been carrie out on the omain size of with a single gri of imensions for the MBCs, an a omain size of with a gri for the full cavity moel. In all moels, 25 gri points are use along the jet exit. These gris are base on the previous simulations of Raju et al. 8,8 which were carrie out over a wie range of flow parameters (Re = , S = 5 5, δ/ = 3, U / = 4), an subjecte to gri refinement an omain epenency stuies. The geometrical an flow parameters in the current work are also chosen base on this previous work. The geometrical parameters for the full cavity are h/ =., W/ = 3. an H/ =.5 (efine in Fig. ). The values of the imensionless flow parameters for the simulations are δ/ = 3., U / = 4, Re = 25 (Re δ = 3), S = 7.7 an St =.2. The incompressible Navier Stokes equations are iscretize using a cell-centre, collocate arrangement of the primitive variables. The solver employs a secon-orer Aams-Bashforth scheme for the convective terms an an implicit Crank-Nicolson scheme for iffusion terms on a Cartesian gris. A secon-orer fractional step metho is applie to solve the equations in time. The solver has been teste extensively by comparisons with the establishe numerical an experimental ata incluing ZNMF jets. 9,2 Canonical Separate Flow Configuration One way to stuy the performance of potential moels is to examine the effect of some of these moels on a separate flow an unerstan to what extent the etails of the actuator moel are necessary to moel the effects of the ZNMF jet on the separation bubble. In orer to aress this issue we have examine a case where a separation bubble is create ownstream of the jet by prescribing a steay blowing-suction bounary conition on the top bounary of the omain. A zero-vorticity conition 2 is applie on this bounary as follows: vx (, L) = Gx ( ), y u y ( x, L y ) = G x (8) where G(x) is the prescribe blowing an suction velocity profile efine in the form of: ( x xc ) Gx ( ) = top sin 2π e L β 2( x xc ) a L (9) where L is the length an x c is the center of the velocity profile. The parameters top, α an β are set to U, an 2, respectively, which form a close separation bubble on the flat plate. A schematic of the flow configuration is shown in Fig. 5. The inflow is a cubic approximation to the Blasius bounary layer u U 3 y y = 2 δ 2 δ 3 () International ournal of Flow Control

10 Ehsan Aram, Rajat Mittal, Louis Cattafesta 7 Slip Conition Suction/Blowing Profile Blasius Inflow Outflow ZNMF Actuator Separation Bubble Non-Slip Conition Figure 5: Flow configuration use for simulating a canonical separate flow. Separate flow simulations have been performe for the following imensionless flow parameters: δ / = 3, U / = 4, Re = 87.5, 25 (Re δ = 225, 3), an three imensionless forcing frequencies F + =.5,.,.5, where F + is the ratio of the jet frequency ( f ) to the ominant frequency in the separation bubble ( f Sep ). We have use a gri on the omain size of for MBC (plug flow), MBC6 an MBC7 moels, an a omain size of with a gri for the FC moel. In all moels, 2 gri points are use across the jet exit. These gris have been use before for ZNMF jet stuies 8,8 an are known to provie aequate solution of the flow. Note that in this component of the stuy, all of the unknowns in the MBC moels are obtaine from FC moel in external flow with no separation bubble (part A). III. RESULTS Moel Performance in icinity of Actuator Base on the criterion suggeste by Holman et al. 7 a strong jet is forme for the current case with St =.2 an it is able to influence the bounary layer at a large istance in the streamwise irection an impart a significant amount of the spanwise vorticity flux to the external flow. Figure 6 shows a comparison of the spanwise vorticity at the peak expulsion for all of the moels at the following conition: Re = 25, S = 7.7, St =.2, U / = 4 an δ / = 3. The strength of the vortex at both eges of the slot exit in MBC (plug flow) is much less than the full cavity moel, but the rest of the moels provies a reasonable qualitative approximation to the full cavity moel. Similar comparisons exist at other phases of the jet cycle x/ x/ x/ x/ 2 4 (FC) (MBC) (MBC2) (MBC3) x/ x/ x/ x/ 2 4 (MBC4) (MBC5) (MBC6) (MBC7) Figure 6: Instantaneous spanwise vorticity uring peak expulsion. olume 2 Number 2 2

11 8 Simple Representations of Zero-Net Mass-Flux ets in Grazing Flow for Flow-Control Simulations The performance of these moels can also be assesse base on the integral quantities such as average flux of spanwise vorticity, u an v momentum an vertical kinetic energy at the jet exit. The efinitions for these quantities are as follows: C uv (t) = ρ u(x, t)v(x, t)x; C vv (t) = ρ u(x, t)v(x, t)x; C vv v(t) = ρ v 2 (x, t)v(x, t)x an Ω(t) = ξ z (x, t) v(x, t)x () where ξ z (x, t) is the spanwise vorticity at the jet exit. The non-imensionalize variation of these quantities over one jet cycle is compare for the various propose moels in Fig. 7. Figure 7: Comparison of the normalize instantaneous (a) u momentum flux, (b) v momentum flux, (c) vertical kinetic energy flux, an () spanwise vorticity flux for case with Re = 25, S = 7.7, St =.2, U / = 4, an δ/ = 3. It is seen that the MBC (plug flow) moel significantly unerpreicts all four quantities uring the expulsion phase, but there is a goo agreement between this moel an the full cavity moel uring the ingestion phase. One reason for this agreement might that the slot acts like a sink uring the ingestion phase an sucks the flow from a semi-infinite meium an using a plug flow profile provies a goo approximation to this flow. MBC2 an MBC3 overpreict the v momentum an vertical kinetic energy flux in the ingestion phase, which means that the two-egree of freeom profile is not a goo caniate uring the ingestion phase. As expecte, the u momentum flux for MBC2 an MBC7 is zero ue to zero u component of velocity in the expulsion phase. It shoul also be note that in general, the horizontal momentum flux is significantly smaller than the flux of vertical momentum. MBC5 has the same performance as MBC4, even though the mean v-velocity has been neglecte in this moel. Comparing the MBC6 an MBC5 cases shows that neglecting the unsteay part of the u-velocity oes not have a significant effect on these quantities an the u momentum flux is only slightly unerpreice. International ournal of Flow Control

12 Ehsan Aram, Rajat Mittal, Louis Cattafesta 9 Table 2: Dimensionless time-average u momentum, v momentum, vertical kinetic energy an spanwise vorticity flux from jet uring the expulsion an ingestion phases. C uv ρ( ) 2 C vv ρ( ) 2 Expulsion Ingestion Expulsion Ingestion Expulsion Ingestion Expulsion Ingestion FC MBC (plug flow) MBC MBC MBC MBC MBC MBC C vvv ρ( ) 3 Ω ( ) 2 Table 2 shows the imensionless time-average of the flux of spanwise vorticity magnitue, u an v momentum flux, an vertical kinetic energy flux at the slot exit uring the expulsion an ingestion phases. As expecte MBC3 (the most complicate moel), MBC4 an MBC5 most accurately preict the mean u momentum flux uring the expulsion phase, whereas MBC (plug flow), MBC2, MBC7 significantly unerpreict this quantity ue to zero u-velocity for these moels in this phase. The mean v momentum an vertical kinetic energy flux of all moels except for MBC have goo agreement with FC moel over the expulsion phase, while MBC2 an MBC3 overpreict these quantities uring the ingestion phase. All the moels except MBC (plug flow) are able to estimate the time average of spanwise vorticity flux over the expulsion phase along the jet exit with reasonable accuracy. Simulations have also been carrie out for five other cases which inclue the following ranges of the parameters: Re = 25 25, S = 5 7.5, St =.33.45, U / = 4 8 an δ / = 3 6 an the results for these are consistent with those for the case escribe in etail in the preceing iscussion. Base on these observations, we select three moels: MBC, MBC6 an MBC7, for further analysis. Note that MBC is the simplest plug flow moel with no unknowns. On the other han, MBC6 an MBC7 are the simplest moels with nonuniform expulsion velocities. Among these two, MBC6 moel has two unknown parameters (U an / (2) ) an MBC7 has only one unknown ( / (2) ) parameter that requires closure. The performance of these selecte moels is examine further by calculating the u momentum, v momentum an vorticity flux within the bounary layer at a ownstream istance of 2 from the actuator. The objective here is to unerstan the extent to which the various moels are able to match the ownstream evolution of the isturbance prouce in the attache bounary by the ZNMF perturbation. In Fig. 8 we show these non-imensionalize integral quantities over the full uration of one expulsion cycle. It is seen that MBC is not able to capture the peak values of these quantities uring the phase where the expelle vortices convect through the region of interest. On the other han, C uu /ru 2 δ φ C uv /ru 2 δ φ φ Figure 8: Comparison of u momentum (left), v momentum (mile) an vorticity flux (right) for selecte moels within the external bounary layer evaluate at 2 ownstream of actuator over one cycle for Re = 87.5, U / = 4, δ/ = 3, S = 7.5, St =.3. Ω/U olume 2 Number 2 2

13 2 Simple Representations of Zero-Net Mass-Flux ets in Grazing Flow for Flow-Control Simulations both MBC6 an MBC7 are able preict both the qualitative as well as quantitative features observe in the temporal variation of these quantities. We therefore expect that these two moels will o a better job in reproucing the effect of the ZNMF jet on a ownstream separation bubble an this issue is examine in the following section. Moel Performance for a Canonical Separate Flow From a practical point-of-view in the context of separation control, the most important measure of fielity of a moel is the extent to which it matches the true effect of the ZNMF jet on a ownstream separation bubble. We have examine this issue by simulating the case escribe in section II.B. Figure 9 shows the unsteay separation bubble forme by applying a steay suction an blowing on the top bounary, ownstream of actuator for two cases Re δ = 225 an Re δ = 3. The averse pressure graient ue to the suction on the top bounary reuces the momentum of the flow an eventually leas to bounary layer separation. The length of the separation bubble base on the mean flow shown by the streamlines in the figure are 4.8 an 4.9 for Re δ = 225 an Re δ = 3, respectively. Figure 9: Instantaneous spanwise vorticity an mean streamlines for the unforce separate flow conition for (a) Re δ = 225 an (b) Re δ = 3. The performance of the selecte moels MBC, MBC6 an MBC7, in this canonical separate flow has been compare with the FC moel in Fig.. The imensionless flow parameters are U / = 4, δ / = 3, Re = 87.5 with forcing frequency F + =.. This figure shows the mean streamlines in the vicinity of the separation bubble for this flow conition. It is clear that MBC moel is not able to capture the seconary separation bubble near the flow etachment point, while the MBC6 an MBC7 moels are able to reprouce this feature. It is also seen that the MBC moel cannot preict the etachment an reattachment of the flow accurately, whereas the location of these points by using MBC6 an MBC7 moels are well matche with the FC moel. A comparison between the maximum separation bubble height for the MBC s moels an FC moel shows that MBC significantly unerpreicts this quantity, whereas the two other moels provie a reasonable preiction of this quantity. One reason for this might be that the v momentum flux for MBC (plug flow) at the jet exit is less than the FC moel uring the expulsion phase. y/ y/ x/ (FC) x/ (MBC6) y/ y/ x/ (MBC) x/ (MBC7) Figure : Mean streamlines of separate flow for U / = 4, δ/ = 3, Re = 87.5, F + =.. International ournal of Flow Control

14 Ehsan Aram, Rajat Mittal, Louis Cattafesta 2 L Sep Reuction% L Sep Reuction% FC MBC MBC6 MBC F + (a) F F + (b) F + Figure : Effect of ifferent forcing frequencies on the separation bubble length an height using FC, MBC, MBC6 an MBC7 moels in (a) Re = 87.5, (b) Re = 25. H Sep Reuction% H Sep Reuction% Figure summarizes the effect of selecte moels on the length an height of separation bubble for three forcing frequencies (F + =.5,.,.5) an two jet Reynols numbers Re = 87.5, 25. It is seen that in low forcing frequencies ( f < f Sep ), where the actuator is effective in reucing the separation bubble, MBC6 an MBC7 moels show goo agreement with the FC moel. On the other han, the MBC unerestimates the separation bubble length an overestimates separation height at low forcing frequencies. Furthermore, whereas MBC6 an MBC7 always correctly preict the tren of separation reuction with F +, MBC actually preicts an incorrect tren of change in L Sep with F + for both cases. Moel Closure The above analysis has ientifie two moels (MBC6 an MBC7) which are relatively simple, an which provie a reasonable improvement in fielity over the plug-flow MBC moel. However, both of these moels have unknown parameters (U an / (2) ) that nee to be etermine a-priori for a given flow configuration in orer to be use in a preictive manner in any simulation. In general, these unknown parameters epen on the external flow as well as jet flow properties: ( 3) ( 2 U /, / ) = f( U /, δ /, S,Re,...) (5) an the above epenence is quite ifficult to etermine a-priori. Here we take a phenomenological approach to moel closure. It is clear that the jet asymmetry (quantifie by / (2) or alternatively R L by ) an slip (U ) shoul epen on the grazing flow velocity upstream of the jet relative / to the jet velocity. A measure of the local grazing flow velocity can be obtaine from the available bounary layer velocity profile upstream of the jet. Alternatively, any simulation with flow control typically has a baseline case with no flow control an this grazing flow velocity coul be extracte from this baseline simulation. While there are many possible ways of efining this velocity, a simple way is to associate this velocity with the streamwise velocity at a istance of olume 2 Number 2 2

15 22 Simple Representations of Zero-Net Mass-Flux ets in Grazing Flow for Flow-Control Simulations from wall upstream of the jet. We enote this velocity as U. Note that for the current Blasius bounary layer an therefore U U 3 = 2 δ 2 δ 3 (6) U U U = 3 f 3 =, δ 2 δ 2 δ (7) Base on our reasoning, ( R / L ) an slip (U ) shoul epen primarily on U / an we check this by extracting ata from a sequence of simulations. Twelve separate ZNMF jet simulations with the following range of parameters have been carrie out: Re = , S = 5, St = , U / = 8 an δ / = 3 6. In aition, we have also carrie out twelve steay jet simulations with these same parameters (these correspon to St = S = ) in orer to examine the extent to which the unsteainess of the jet impact the above scaling. The time average of the unknown parameters evaluate from the steay an unsteay jet for ifferent U / an δ /, an fixe Re an St number are compare in Fig. 2. Interestingly, there is a fairly goo agreement between the ZNMF an steay jets inicating that the physical processes that lea to jet asymmetry an slip are not intrinsically unsteay. U o / Unsteay et, Extracte Data Steay et, Extracte Data.5 Unsteay et, Regression Steay et, Regression U / U / (a) (b) Figure 2: Comparison between the steay an unsteay jet for time average of (a) u-velocity an (b) v-velocity ratio along the slot exit for ifferent U / an δ/. Re an St are fixe to 87.5 an.3 respectively. R L / We now use regression (least-squares fit) to etermine an empirical power-law epenence of asymmetry an slip on U /. In conucting the regression analysis, we have also impose the following limiting conition that applies for a jet in quiescent flow: U R U lim & lim / U / L (8) The regression leas to the following expression for the steay an unsteay jets respectively: U 45 R U = 3. U. ; = +. L 6. (9) International ournal of Flow Control

16 Ehsan Aram, Rajat Mittal, Louis Cattafesta R U U = U. ; = +. (2) L It is also of interest to examine the epenence of the jet asymmetry an slip on the jet Strouhal an Reynols numbers. Figure 3(a) compares the mean value of these two parameters for steay an unsteay jet for three ifferent St an fixe U /, δ/ an Re. It is clear from this plot that both values are fairly inepenent of the Strouhal number in the range of stuy. Figure 3(b) shows the effect of Re number on the unknown parameters for steay an unsteay jet where the rest of flow parameters are fixe. It is seen that U is nearly inepenent of jet Reynols number for both steay an unsteay jets whereas R / L has a weak epenence on the Reynols number. Thus, base on the above analysis we propose the following general forms for the scaling of the jet slip an asymmetry parameters: U R L A U = While we have foun values for A, B, β an γ for our generic configuration, it is likely that there is some variation in these values for each ifferent configuration. Furthermore, turbulence in the grazing bounary layer might also moify the epenence, since ingestion of a turbulent flow by the actuator might change the flow pattern insie the slot an cavity an eventually of the jet coming out of the slot. Nevertheless, given that the above scaling laws seem to apply to unsteay as well as steay jets implies that the scaling approach is quite robust in representing the essential flow physics an woul therefore be wiely applicable. Furthermore, the approach escribe here might be irectly applicable for the mean flow component of a turbulent flow as in a Reynols-average Navier-Stokes (RANS) simulation. β B U = γ κ Re (2) Figure 3: Comparison between the steay an unsteay jet for time average of u-velocity an v-velocity ratio along the slot exit for (a) ifferent St number, fixe U / = 4, δ/ = 3 an Re = 87.5 an (b) ifferent Re number, fixe U / = 4, δ/ = 3, an St =.3. I. CONCLUSIONS A class of moifie bounary conition moels is propose to reprouce the flow associate with ZNMF jets in grazing flows. The propose moels have a nonuniform jet velocity profile with only two spatial egrees of freeom an a uniform slip velocity on the slot-flow bounary. A comparison of key integral quantities associate with momentum, energy an vorticity flux shows that the moels with nonuniform jet velocity uring the expulsion phase an uniform jet velocity uring the ingestion phase can preict these quantities with goo accuracy, whereas a simple plug flow moel with zero slip an uniform jet velocity unerpreicts these three quantities uring the expulsion phase. olume 2 Number 2 2

17 24 Simple Representations of Zero-Net Mass-Flux ets in Grazing Flow for Flow-Control Simulations Base on our initial analysis, three of the simplest moels were selecte for further stuy, incluing an assessment of their performance for a canonical separate flow at ifferent forcing frequencies. A key fining of this stuy is that a simple plug-flow type moel can preict incorrect trens for separation reuction with jet frequency. Thus caution nees to be exercise in using such simple moels in computational stuies of flow control. It is also foun that inclusion of a jet flow asymmetry parameter improves the fielity of the moel significantly an leas to better qualitative an quantitative preiction of separation control. Inclusion of a non-zero slip velocity improves the preiction of horizontal momentum flux from the jet but oes not significantly improve the fielity of the moel. A preliminary attempt has been mae to provie empirical closure to the above moels. Our simulations suggest that the two parameters in the moels: jet asymmetry an slip, are well represente through a power law epenence on a parameter that quantifies the grazing flow velocity of the bounary layer upstream of the jet relative to the jet velocity. This parameter can be etermine concurrently from a flow control simulation an the jet asymmetry moel ajuste uring the course of the simulation. Alternatively this quantity might be available from a baseline flow simulation (with no flow control) or from a potential flow type moel of the configuration uner consieration. The jet asymmetry also shows a weak power law epenence on the jet Reynols number. Further analysis of these scaling laws an the extent to which they are universal nees to be investigates through aitional analysis an simulations. ACKNOWLEDGEMENTS The work presente here is supporte by NASA uner cooperative agreement NNX7AD94A, monitore by Dr. Brian Allan. Partial support from AFOSR grant FA is also acknowlege. REFERENCES. Seifert, A., Bachar, T., Koss, D., Shepshelovich, M., an Wygnanski, I., Oscillatory Blowing: A Tool to Delay Bounary-Layer Separation, AIAA ournal, ol. 3, No., 993, pp Wang, H., an Menon, S., Fuel-Air Mixing Enhancement by Synthetic Microjets, AIAA ournal, ol. 39, No. 2, 2, pp Mahalingam, R., an Glezer, A., Design an Thermal Characteristics of a Synthetic et Ejector Heat Sink, ournal of Electronic Packaging, ol. 27(2), 25, pp Smith, B. L., an Glezer, A., et ectoring using Synthetic ets, ournal of Flui Mechanics, ol. 458, 22, pp Gallas, Q., Holman, R., Nishia, T., Carroll, B., Sheplak, M., an Cattafesta, L., Lumpe Element Moeling of Piezoelectric-Driven Synthetic et Actuators, AIAA ournal, ol. 4, No. 2, 23, pp Glezer, A., an Amitay, M., Synthetic ets, Review of Flui Mechanics, ol. 34, 22, pp Holman, R., Utturkar, Y., Mittal, R., Smith, B. L., an Cattafesta, L., Formation Criterion for Synthetic ets, AIAA ournal, ol. 43, No., 25, pp Raju, R., Aram, E., Mittal, R., an Cattafesta, L. N., Simple Moels of Zero-Net Mass-Flux ets for Flow Control Simulations, International ournal of Flow Control, ol., No. 3, Kral, L. D., Donovan,. F., Cain, A. B., an Cary, A. W., Numerical Simulation of Synthetic et Actuators, AIAA Paper , Rizzetta, D. P., isbal, M. R., an Stanek, M.., Numerical Investigation of Synthetic-et Flow Fiels, AIAA ournal, ol. 37, No. 8, 999, pp Filz, C., Lee, D., Orkwis, P. D., an Turner, M. G., Moeling of Two Dimensional irecte Synthetic ets using Neural Network-base Detereministic Source Terms, AIAA Paper , Yamaleev, N. K., an Carpenter, M. H., A Reuce-Orer Moel for Efficient Simulation of Synthetic et Actuators, NASA Technical Reports, NASA/TM , Rathnasingham, R. an Breuer, K. S., Couple Flui-Structural Characteristics of Actuators for Flow Control, AIAA ournal, ol. 35, No. 5, 997, pp International ournal of Flow Control

18 Ehsan Aram, Rajat Mittal, Louis Cattafesta Agashe,., Arnol, D. P., an Cattafesta, L., Lumpe Element Moel for Electroynamic Zero- Net Mass-Flux Actuators, AIAA Paper 29-38, anuary Byrganhalli, R. R., Numerical Stuy of Three Dimensional Synthetic ets in Quiescent an External Grazing Flows, D.Sc. Thesis, 27, Mechanical an Aerospace Engineering, The George Washington University, Washington, DC. 6. Tang, H., Zhong, S., abbal, M., Garcillan, L., Guo, F., Woo, N.., an Warsop, C., Towars the Design of Synthetic-et Actuators for Full-Scale Flight Conitions. Part 2: Low-imensional Atuator Preiction Moels an Actuator Design Methos, Flow, Turbulence an Combustion, ol. 78, No. 3, 27, pp Utturkar, Y., Mittal, R., Rampungoon, P., an Cattafesta, L., Sensitivity of Synthetic ets to the Design of the et Cavity, AIAA Paper Raju, R., Scaling Laws an Separation Control Strategies for Zero-Net Mass-Flux Actuators, D.Sc. Thesis, 28, Mechanical an Aerospace Engineering, The George Washington University, Washington, DC. 9. Kotapati, R. B., Mittal, R., an Cattafesta, L. N., Numerical Stuy of Transitional Synthetic et in Quiescent External Flow, ournal of Flui Mechanics, ol. 58, 27, pp Mittal, R., Dong, H., Bozkurttas, M., Najjar, F., argas, A., an Loebbecke, A., A ersatile Immerse Bounary Metho for Incompressible Flows with Complex Bounaries, ournal of Computational Physics, ol. 227, No., 28, pp Na, Y., an Moin, P., Direct Numerical Simulation of a Separate Turbulent Bounary Layer, ournal of Flui Mechanics, ol. 37, 998, pp olume 2 Number 2 2

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